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Semirings

Louis Halle Rowen

Abstract

We survey theory developed over the past 10 years of semirings which need not be additively cancellative. The main feature is a specified ``null ideal'' $\mcA_0$ of a semiring $\mcA,$ taking the place of a zero element, which permits generalizations of the classical algebraic theory to polynomials and their roots, algebraic geometry, matrices, linear algebra, varieties, categories, and module theory. The ``pair'' $(\mcA,\mcA_0)$ is studied along the lines of universal algebra.

Semirings

Abstract

We survey theory developed over the past 10 years of semirings which need not be additively cancellative. The main feature is a specified ``null ideal'' of a semiring taking the place of a zero element, which permits generalizations of the classical algebraic theory to polynomials and their roots, algebraic geometry, matrices, linear algebra, varieties, categories, and module theory. The ``pair'' is studied along the lines of universal algebra.
Paper Structure (50 sections, 40 theorems, 16 equations)

This paper contains 50 sections, 40 theorems, 16 equations.

Table of Contents

  1. Overview
  2. Additively cancellative semirings
  3. Idempotent semirings
  4. The max-plus algebra
  5. Connection to tropical geometry
  6. Valuations and the supertropical semiring
  7. The supertropical semiring
  8. Tropical extensions
  9. Graded semirings
  10. Hyperrings and Hyperfields
  11. Examples of hyperfields
  12. Krasner's residue construction
  13. Description of major hyperfields as quotient hyperfields
  14. Semiring constructions
  15. Doubling
  16. ...and 35 more sections

Key Result

Lemma 1.11

$$

Theorems & Definitions (147)

  • Definition 1.2
  • Example 1.3
  • Remark 1.4
  • Definition 1.6
  • Remark 1.7
  • Definition 1.8
  • Definition 1.9
  • Definition 1.10
  • Lemma 1.11
  • proof
  • ...and 137 more